class: center, middle, inverse, title-slide .title[ # Beyond simple maps - Integrating space and time with Bayesian models ] .subtitle[ ## Summer at Census Research Seminar ] .author[ ### Corey S. Sparks, Ph.D. ] .institute[ ### Univerity of Texas at San Antonio - Department of Demography ] .institute[ ### <a href="https://hcap.utsa.edu/demography" class="uri">https://hcap.utsa.edu/demography</a> ] .date[ ### July 11, 2022 ] --- ## Presentation Structure - Spatial and temporal demography - Data sources - Modeling strategies - Empirical analysis of Florida mortality rates - Results & visualizations - Wrap up --- ##Beyond maps... .center[] --- class: center, inverse .center[] --- class: center, inverse .center[] --- ## Spatial Demography - "Putting people into place" (Entwisle, 2007) + Need to think about: + Context + Dynamics + Processes -- - Macro - demography (Voss, 2007) + Places as observations + Pre - 1960's + Ecological inference -- - Micro - demography + People as observations + Social theory + Individual choices -- - Multilevel - demography + People in places + Interaction between context and behavior --- ## Space & Time - [Future directions in spatial demography](https://escholarship.org/uc/item/7xx3k2z4) report + Most participants listed time or temporal data as integral to the future of the field -- - Time allows for dynamics of humans and environment + Snap shots/cross sections tell us nothing of this --- class: center, middle ## Space & Time data models  [Kisilevich et al 2010](https://link-springer-com.libweb.lib.utsa.edu/chapter/10.1007/978-0-387-09823-4_44) --- <style type="text/css"> .pull-left { float: left; width: 55%; } .pull-right { float: right; width: 44%; } .pull-right ~ p { clear: both; } </style> ## Complexities .pull-left[ - Humans, I mean c'mon <img src="hurricane.jpg" width="350px" style="display: block; margin: auto;" /><img src="border.jpg" width="350px" style="display: block; margin: auto;" /> ![]() ] .pull-right[ <img src="hwy.jpg" width="350px" style="display: block; margin: auto;" /> ![]() ] --- ## Complexities - Data sources ? + Surveys + __Administrative data__ -- - Data management + Combining and merging data -- - Analysis/methods + Problems with space + Problems with time -- - Advantages + Rich, dynamic contexts + Policy relevance of timely, prospective analysis --- ## Data sources - NCHS/CDC -- - Census/ACS -- - DHS -- - IPUMS -- - International agencies -- - Various administrative orgs. + State government + Private companies/Nonprofits --- ## How to combine these things? - Geocodes are essential + Limitation for many surveys -- - **Caveats** - Levels of geography + The evil tracts - MAUP - Changing boundaries - Analytically + Lots of ways, but are they all ideal? + These data can often be *very* large in size --- ## Hierarchical Models - Allow for nesting of individuals by many different levels + People within places, within time periods - Different types of outcomes + Continuous/discrete observations/outcomes - Can include correlation between higher level units + Autocorrelation between places/time periods - Dynamic modeling + Place - specific time trends for example --- ## Empirical example - US County Mortality Rates - NCHS [Compressed Mortality File](http://www.cdc.gov/nchs/data_access/cmf.htm) + County - level counts of deaths by year, age, sex, race/ethnicity and cause of death + 1980 to 2010 + Age, sex and race _(white & black)_ specific rates for all US counties + In total: 35,748,276 deaths in the data + Standardized to 2000 Standard US population age structure + Rates stratified by race and sex for each county by year + n = 2 sexes `\(*\)` 2 races `\(*\)` 3106 counties `\(*\)` 31 years = 385,144 observations + *Analytic* n = 315,808 nonzero rates -- - You can basically get these data from the CDC Wonder [website](http://wonder.cdc.gov/mortsql.html) - Suppresses counts where the number of deaths is less than 10 - Rates are labeled as "__unreliable__" when the rate is calculated with a numerator of 20 or less + Big problem for small population counties + Still a problem for large population counties! -- - Restricted use data allows access to __ALL__ data --- ## Data example | County | Year | Race-Sex | Rate | |:------:|:----:|:------------:|:---------:| | 12073 | 1980 | White Female | 7.238632 | | 12073 | 1980 | Black Female | 8.958174 | | 12073 | 1980 | White Male | 11.840842 | | 12073 | 1980 | Black Male | 15.907688 | | 12073 | 1981 | White Female | 7.383039 | | 12073 | 1981 | Black Female | 9.379846 | | 12073 | 1981 | White Male | 10.518428 | | 12073 | 1981 | Black Male | 16.626825 | | 12073 | 1982 | White Female | 7.370335 | | 12073 | 1982 | Black Female | 8.695655 | | 12073 | 1982 | White Male | 11.902308 | | 12073 | 1982 | Black Male | 12.149819 | --- County specific temporal trends 1980 - 2010 <!-- --> --- ## Florida Example - n = 67 counties `\(*\)` 31 years `\(*\)` 2 Races `\(*\)` 2 Sexes = 8,308  --- ## Methods - Bayesian Hierarchical models * Example case of Florida counties * Examine county-specific time trends in Black/White mortality rates * I specify a Bayesian Hierarchical model for the age-standardized mortality rate * Controls for sex and county SES * Spatial correlation in overall rate `\(u_j\)` * Time varying Black/white disparity parameter `\(\nu_{t2}\)` * Spatially varying Black/White disparity parameter `\(\gamma_j\)` $$ `\begin{aligned} \operatorname{y}_{ij} &\sim N\left( \mu, \tau_y \right) \\ & \mu_{ij} = \beta_{0} + x'\beta +\gamma_j*Black + u_j +\nu_{t1} + \nu_{t2}* Black \\ & \gamma_j \sim \text{CAR}(\bar \gamma_j, \tau_{\gamma}/n_j) \\ & u_j \sim \text{CAR}(\bar u_j, \tau_u /n_j)\\ & \nu_{t2} \sim RW1(time)\\ & \nu_{t1} \sim N(0, \tau_t) \\ \end{aligned}` $$ --- ## Methods - Bayesian Hierearchical models * This type of model is commonly used in epidemiology and public health * Various types of data likelihoods may be used * Need to get at: *$$p(\theta|y) \propto p(y|\theta)p(\theta)$$ * Traditionally, we would get `\(p(\theta|y)\)` by: + either figuring out what the full conditionals for all our model parameters are (hard) + Use some form of MCMC to arrive at the posterior marginal distributions for our parameters (time consuming) --- ## Methods - INLA approach * [Integrated Nested Laplace Approximation](http://www.math.ntnu.no/~hrue/r-inla.org/papers/inla-rss.pdf) - Rue, Martino & Chopin (2009) * One of several techniques that approximate the marginal and conditional posterior densities + Laplace, PQL, E-M, Variational Bayes * Assumes all random effects in the model are latent, zero-mean Gaussian random field, `\(x\)` with some precision matrix + The precision matrix depends on a small set of hyperparameters * Attempts to construct a joint Gaussian approximation for `\(p(x | \theta, y)\)` + where `\(\theta\)` is a small subset of hyper-parameters --- ## Methods - INLA approach * Apply these approximations to arrive at: * `\(\tilde{\pi}(x_i | y) = \int \tilde{\pi}(x_i |\theta, y)\tilde{\pi}(\theta| y) d\theta\)` * `\(\tilde{\pi}(\theta_j | y) = \int \tilde{\pi}(\theta| y) d\theta_{-j}\)` * where each `\(\tilde{\pi}(. |.)\)` is an approximated conditional density of its parameters * Approximations to `\(\pi(x_i | y)\)` are computed by approximating both `\(\pi(\theta| y)\)` and `\(\pi(x_i| \theta, y)\)` using numerical integration to integrate out the nuisance parameters. + This is possible if the dimension of `\(\theta\)` is small. * Approximations to `\(\tilde{\pi}(\theta|y)\)` are based on the Laplace appoximation of the marginal posterior density for `\(\pi(x,\theta|y)\)` * Their approach relies on numerical integration of the posterior of the latent field, as opposed to a pure Gaussian approximation of it --- ## INLA in R `library(INLA)` `std_rate~male+black+scale(lths)+` `f(year2, model = "rw1",constr = T, scale.model = T)+` **nonparametric time trend** `f(struct, model="besag", graph="cl_graph", constr = T, scale.model = T)+` **spatial correlation** `f(year3, bl2, model="iid")+` **time - disparity** `f(struct2, bl2, model="besag", graph="cl_graph", constr = T, scale.model = T)` **spatial disparity** --- ## Results - Time trend in Black/white Mortality <!-- --> --- ## County time trends <!-- --> --- ## Highlighed trends <!-- --> --- ## Spatial trend <!-- --> --- ## Spatial disparity <!-- --> --- ## Discussion * We see that, while there is a persistence of the gap in black-white mortality: + The mortality gap appears to be fairly consistent over time + In some areas, the mortality difference are decreasing + Results point to higher disparities in several notable Florida rural areas * Spatio-temporal modeling allows for the incorporation of dynamics that cross-sectional models cannot --- ## Low Response Score Outcome - INLA model for Low Response Score metric - Considered both an unstructured and spatially structured random effect model - Modeled LRS as Gaussian, considering how it is constructed - Besag, York and Mollie specification for tract level heterogeneity $$ `\begin{aligned} \operatorname{y}_{i} &\sim \text{Normal}\left( \mu_i, \tau_y \right) \\ & \mu_{i} = \beta_{0} + u_i + v_i \\ & u_i \sim \text{CAR}(\bar u_i, \tau_{u}/n_j) \\ &